A003132 -id:A003132 - cp's OEIS frontend

A048377 Append d digits d after each digit d in decimal expansion of n.

0, 11, 222, 3333, 44444, 555555, 6666666, 77777777, 888888888, 9999999999, 110, 1111, 11222, 113333, 1144444, 11555555, 116666666, 1177777777, 11888888888, 119999999999, 2220, 22211, 222222, 2223333, 22244444, 222555555, 2226666666

Offset: 0

Patrick De Geest, Mar 15 1999

a(10^n) = 11 * 10^n; A007953(a(n)) = A003132(n) + A007953(n). [Reinhard Zumkeller, Jul 10 2011]

			12 becomes 1 1 2 22 = 11222.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A048376.

import Data.Char (digitToInt)
a048377 :: Integer -> Integer
a048377 n =
   read $ concat $ zipWith replicate (map ((+ 1) . digitToInt) ns) ns
      where ns = show n
-- Reinhard Zumkeller, Jul 10 2011

den[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Flatten[Table[ Table[ idn[[i]],{idn[[i]]+1}],{i,Length[idn]}]]]]; Array[den,30,0] (* Harvey P. Dale, Sep 04 2013 *)

A094830 Start with x = n, repeatedly replace x with x + sum of squares of digits of x until you reach a prime; sequence gives number of steps.

Original entry on oeis.org

1, 0, 0, 6, 0, 4, 0, 11, 5, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 5, 13, 9, 0, 4, 11, 12, 17, 3, 0, 8, 0, 8, 7, 1, 7, 3, 0, 5, 7, 4, 0, 3, 0, 3, 7, 8, 0, 2, 2, 2, 6, 3, 0, 10, 2, 3, 1, 3, 0, 3, 0, 2, 2, 18, 2, 11, 0, 2, 6, 9, 0, 10, 0, 1, 1, 2, 5, 1, 0, 16, 2, 8, 0, 3, 11, 6, 10, 2, 0, 4, 1, 15, 2, 1, 9, 2, 0, 7, 7, 1

Offset: 1

Jason Earls, Jun 13 2004

It is currently not known if this sequence is well defined, i.e. if for all n>=1 there exists an integer v such that a(n) = v. If the sequence is not well defined then the given programs are incorrect as they will get stuck in an infinite loop for some integers. - Peter Luschny, Oct 28 2016

			a(4)=6 because 4 -> 20 -> 24 -> 44 -> 76 -> 161 -> 199, takes 6 steps to reach a prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003132, A094877.

f:= proc(n) local x,k;
x:= n;
for k from 0 do
  if isprime(x) then return k fi;
  x:= x + add(t^2, t = convert(x,base,10))
od;
end proc:
map(f, [$1..100]); # Robert Israel, Oct 27 2016

p[n_]:=Length[NestWhileList[#+Total[IntegerDigits[#]^2]&,n, !PrimeQ[ #]&]]-1; Array[p,100] (* Harvey P. Dale, Dec 03 2011 *)

a(n) = my(k=0); while(!isprime(n),d=digits(n); n+=vecsum(vector(#d,i,d[i]^2)); k++) ;k \\ David A. Corneth, Oct 27 2016

A033936 a(n+1) = a(n) + sum of squares of digits of a(n).

Original entry on oeis.org

1, 2, 6, 42, 62, 102, 107, 157, 232, 249, 350, 384, 473, 547, 637, 731, 790, 920, 1005, 1031, 1042, 1063, 1109, 1192, 1279, 1414, 1448, 1545, 1612, 1654, 1732, 1795, 1951, 2059, 2169, 2291, 2381, 2459, 2585, 2703, 2765, 2879, 3077, 3184

Offset: 0

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Orbit of 1 under iterations of A258881. - M. F. Hasler, Jul 23 2015

			After 1063, since 1^2 + 0^2 + 6^2 + 3^2 = 46 we get 1063+46 = 1109.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A258881, A003132, A076161.

A[0] := 1;
for n to 50 do
   A[n] := A[n-1]+add(t^2, t = convert(A[n-1], base, 10))
od:
seq(A[i],i=0..50); # Robert Israel, Feb 15 2018

NestList[#+Total[IntegerDigits[#]^2]&,1,45] (* Harvey P. Dale, Aug 14 2011 *)

a(n+1) = A258881(a(n)). a(n) = (A258881^n)(1). - M. F. Hasler, Jul 23 2015

A055016 Smallest number whose sum of squares of digits is n.

Original entry on oeis.org

0, 1, 11, 111, 2, 12, 112, 1112, 22, 3, 13, 113, 222, 23, 123, 1123, 4, 14, 33, 133, 24, 124, 233, 1233, 224, 5, 15, 115, 1115, 25, 125, 1125, 44, 144, 35, 135, 6, 16, 116, 1116, 26, 45, 145, 335, 226, 36, 136, 1136, 444, 7, 17, 117, 46, 27, 127, 1127, 246

Offset: 0

Henry Bottomley, May 31 2000

Zak Seidov, Table of n, a(n) for n = 0..600
E. Angelini, Sum of squares -- and a concatenation, SeqFan list, June 23, 2015.

Cf. A003132 is inverse in sense that n=A003132(a(n)), though not necessarily a(A003132(n)).

A055016(n,q=9,m)={ n>1||return(n); forstep(q=min(sqrtint(n),q),1,-1, m && n \ q^2 * #Str(q) > #Str(m) && break; t=eval(Str(A055016(n-q^2,q),q)); (!m || tM. F. Hasler, Jun 24 2015

A069816 a(n) = (sum of digits of n)^2 - (sum of digits^2 of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 0, 12, 24, 36, 48, 60, 72, 84, 96, 108, 0, 14, 28, 42, 56, 70, 84

Offset: 0

N. J. A. Sloane, May 03 2002

a(n) = 0 iff n is a single-digit number or is a power of ten times a single-digit number. - Robert G. Wilson v, May 04 2002

			n=14: sum of digits = 5, squaring gives 25; sum of squared digits of n = 1^2 + 4^2 = 17, so a(14) = 25 - 17 = 8.

Cf. A067552.

A007953 := proc(n) add(d, d=convert(n,base,10)) ; end proc: A003132:= proc(n) add(d^2, d=convert(n,base,10)) ; end proc: A118881 := proc(n) (A007953(n))^2 ; end proc: A069816 := proc(n) A118881(n)-A003132(n) ; end proc: seq(A069816(n),n=0..80) ; # R. J. Mathar, Jun 23 2009

f[n_] := Plus @@ IntegerDigits[n]^2 - Plus @@ (IntegerDigits[n]^2); Table[ f[n], {n, 0, 100}]

From R. J. Mathar, Jun 23 2009: (Start)
a(n) = 2*A035930(n), n <= 100.
a(n) = A118881(n) - A003132(n). (End)

Extended by Robert G. Wilson v, May 04 2002

A090425 Number of iterations required for happy number A007770(n) to converge to 1.

Original entry on oeis.org

1, 6, 2, 3, 5, 4, 4, 3, 4, 5, 5, 3, 6, 4, 4, 3, 5, 5, 4, 2, 3, 5, 4, 3, 6, 6, 4, 4, 5, 5, 4, 6, 4, 4, 4, 6, 4, 6, 6, 6, 6, 4, 4, 6, 3, 4, 3, 6, 6, 4, 6, 6, 6, 5, 7, 6, 7, 6, 6, 6, 7, 5, 6, 6, 6, 7, 5, 5, 5, 4, 4, 7, 5, 5, 5, 7, 7, 4, 7, 4, 5, 3, 4, 6, 6, 6, 7, 6, 6, 4, 7, 7, 4, 5, 5, 4, 6, 3, 6, 7, 6, 4

Offset: 1

Eric W. Weisstein, Nov 30 2003

The count includes both the start and end.

			7 is the 2nd happy number and iterated digit squarings and additions give the sequence {7,49,97,130,10,1}, so a(2)=6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Cai and Xia Zhou, On the heights of happy numbers, Rocky Mount. J. Math. 38 (6) (2008) 1921-1926. [From _R. J. Mathar_, Apr 22 2010]
Eric Weisstein's World of Mathematics, Happy Number.

Cf. A007770.

Cf. A003132.

a090425 n = snd $ until ((== 1) . fst)
                        (\(u, v) -> (a003132 u, v + 1)) (a007770 n, 1)
-- Reinhard Zumkeller, Aug 07 2012

happy[n_] := If[(list = NestWhileList[Plus @@ (IntegerDigits[#]^2) &, n, UnsameQ, All])[[-1]] == 1, Length[list] - 1, Nothing]; Array[happy, 700] (* Amiram Eldar, Apr 12 2022 *)

from itertools import count, islice
def A090425_gen(): # generator of terms
    for n in count(1):
        c = 1
        while n not in {1,37,58,89,145,42,20,4,16}:
            n = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(n))
            c += 1
        if n == 1:
            yield c
A090425_list = list(islice(A090425_gen(),20)) # Chai Wah Wu, Aug 02 2023

A103369 Number in the 2-digitaddition sequence at which the eventually periodic part starts.

Original entry on oeis.org

1, 4, 37, 4, 89, 89, 1, 89, 37, 1, 4, 89, 1, 89, 16, 16, 89, 37, 1, 20, 89, 89, 1, 20, 89, 16, 89, 1, 89, 37, 1, 1, 37, 89, 89, 89, 37, 58, 37, 16, 89, 42, 89, 1, 89, 89, 37, 89, 1, 89, 16, 89, 89, 89, 89, 37, 37, 58, 37, 89, 37, 16, 89, 89, 37, 89, 89, 1, 16, 1, 89, 89, 58

Offset: 1

Eric W. Weisstein, Feb 02 2005

a(A007770(n)) = 1; a(A031177(n)) > 1. - Reinhard Zumkeller, Mar 16 2013

			The 2-digitaddition sequence for n = 3 is {3, 9, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, ...}, so a(3) = 37.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digitaddition

Cf. A007770, A031176, A003132, A039943.

a103369 = until (`elem` a039943_list) a003132
a103369_list = map a103369 [1..]
-- Reinhard Zumkeller, Oct 17 2011, Aug 24 2011

A226224 The largest value of k in base n for which the sum of digits of k = sqrt(k).

Original entry on oeis.org

1, 25, 9, 64, 100, 144, 49, 64, 81, 225, 121, 441, 169, 441, 441, 256, 289, 324, 361, 1296, 1296, 484, 529, 1089, 625, 676, 729, 2401, 841, 2601, 961, 1024, 3025, 1156, 2500, 4096, 1369, 1444, 4356, 3136, 1681, 4900, 1849, 5929, 3025, 2116, 2209, 6561, 2401

Offset: 2

Christian N. K. Anderson and Kevin L. Schwartz, May 31 2013

There are no values of k in base n with more than 3 digits. Proof: such a value with d digits would need to meet the criterion d*(n-1)>=sqrt(n)^d which establishes an upper limit of 4 digits for 2<=n<=6 and 3 for n>6. Because there are no four digit values of k in bases 2 through 6, k has a maximum of three digits in all bases.
Because k must be a square, there are only sqrt(n)^3 possible values in any base.
From the above, it can be shown that for three-digit fixed points of the form xyz, x <= 6; also x<=4 for n>846. These theoretical upper limits are statistically unlikely, and in fact of the 86356 solutions in bases 2 to 10000, only 6.5% of them begin with 2, and none begin with 3 through 6.
a(n)=1 iff A226087(n)=1. Conjecture: this occurs exactly once -- in base 2.

			For a(16) the solutions are the square numbers {1, 36=6^2, 100=10^2, 225=15^2, 441=21^2} because in base 16 they are written as {1, 24, 64, E1, 1B9} and 1 = 1, 6 = 2+4, 10 = 6+4, 15 = 14+1, and 21 = 1+11+9.

Christian N. K. Anderson, Table of n, a(n) for n = 2..10000
Christian N. K. Anderson, Table of base, max(k), number of k, and all values of k for base 2 to 10000.

Cf. A007953, A003132, A055012, A055013, A055014, A055015, A226087.

for(n in 2:500) cat("Base",n,":",which(sapply((1:(ifelse(n>6,7,1)*n^ifelse(n>6,1,2)))^2, function(x) sum(inbase(x,n))==sqrt(x)))^2, "\n")

A052035 Palindromic primes whose sum of squared digits is also prime.

Original entry on oeis.org

11, 101, 131, 191, 313, 353, 373, 797, 919, 10301, 11311, 12721, 13331, 13931, 14341, 14741, 16361, 17971, 18181, 19391, 30103, 30703, 33533, 71317, 71917, 74747, 75557, 76367, 77977, 79397, 90709, 93139, 93739, 95959, 96769, 97379

Offset: 1

Patrick De Geest, Dec 15 1999

From Bernard Schott, Oct 20 2021: (Start)
Except for 11, all terms have an odd number of digits.
Except for terms of the form 10^k+1, k >= 2, the middle digit is always odd; the unique known term of the form 10^k+1 for 2 <= k <= 100000 is 101 (see comment in A000533). (End)

			373 -> 3^2 + 7^2 + 3^2 = 67, which is prime.

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Michel Marcus, Table of n, a(n) for n = 1..1215
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A000533, A002385, A052034, A003132.

Select[Prime@ Range[2, 10^4], And[PalindromeQ@ #, PrimeQ@ Total[IntegerDigits[#]^2]] &] (* Michael De Vlieger, Oct 20 2021 *)

isok(p) = my(d=digits(p)); isprime(p) && (d==Vecrev(d)) && isprime(sum(k=1, #d, d[k]^2)); \\ Michel Marcus, Oct 17 2021

from sympy import isprime
def ok(n):
    s = str(n)
    return s==s[::-1] and isprime(n) and isprime(sum(int(d)**2 for d in s))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 23 2021

# second version for going to large terms
from sympy import isprime
from itertools import product
def ok(pal):
    return isprime(pal) and isprime(sum(int(d)**2 for d in str(pal)))
def agentod(maxdigs):
    yield 11
    for d in range(3, maxdigs+1, 2):
        pal = 10**(d-1) + 1
        if ok(pal): yield pal
        for first in "1379":
            for left in product("0123456789", repeat=(d-3)//2):
                left = "".join(left)
                for mid in "13579":
                    pal = int(first + left + mid + left[::-1] + first)
                    if ok(pal): yield pal
print([an for an in agentod(5)]) # Michael S. Branicky, Nov 23 2021

A123253 Sum of 7th powers of digits of n.

Original entry on oeis.org

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 128, 129, 256, 2315, 16512, 78253, 280064, 823671, 2097280, 4783097, 2187, 2188, 2315, 4374, 18571, 80312, 282123

Offset: 0

Zerinvary Lajos, Nov 06 2006

Fixed points are listed in A124068 = row n=7 of A252648. - M. F. Hasler, Apr 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003132, A055012, A055013, A055014, A055015, A210840.

[0] cat [&+[d^7: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

A123253 := proc(n)
        add(d^7,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jan 16 2013

Table[Sum[DigitCount[n][[i]] i^7, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A123253(n)=sum(i=1,#n=digits(n),n[i]^7) \\ M. F. Hasler, Apr 12 2015

cp's OEIS Frontend

A048377 Append d digits d after each digit d in decimal expansion of n.

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A094830 Start with x = n, repeatedly replace x with x + sum of squares of digits of x until you reach a prime; sequence gives number of steps.

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A033936 a(n+1) = a(n) + sum of squares of digits of a(n).

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A055016 Smallest number whose sum of squares of digits is n.

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A069816 a(n) = (sum of digits of n)^2 - (sum of digits^2 of n).

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A090425 Number of iterations required for happy number A007770(n) to converge to 1.

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A103369 Number in the 2-digitaddition sequence at which the eventually periodic part starts.

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A226224 The largest value of k in base n for which the sum of digits of k = sqrt(k).

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